Enriched algebraic theories and monads for a system of arities

Under a minimum of assumptions, we develop in generality the basic theory of universal algebra in a symmetric monoidal closed category V with respect to a specified system of arities j : J ↪→ V . Lawvere’s notion of algebraic theory generalizes to this context, resulting in the notion of single-sorted V -enriched J -cotensor theory, or J -theory for short. For suitable choices of V and J , such J -theories include the enriched algebraic theories of Borceux and Day, the enriched Lawvere theories of Power, the equational theories of Linton’s 1965 work, and the V -theories of Dubuc, which are recovered by taking J = V and correspond to arbitrary V -monads on V . We identify a modest condition on j that entails that the V -category of T -algebras exists and is monadic over V for every J -theory T , even when T is not small and V is neither complete nor cocomplete. We show that j satisfies this condition if and only if j presents V as a free cocompletion of J with respect to the weights for left Kan extensions along j, and so we call such systems of arities eleutheric. We show that J -theories for an eleutheric system may be equivalently described as (i) monads in a certain one-object bicategory of profunctors on J , and (ii) V -monads on V satisfying a certain condition. We prove a characterization theorem for the categories of algebras of J -theories, considered as V -categories A equipped with a specified V -functor A → V .